Circuit quantization in the presence of time-dependent external flux

Circuit quantization links a physical circuit to its corresponding quantum Hamiltonian. The standard quantization procedure generally assumes any external magnetic flux to be static. Time dependence naturally arises, however, when flux is modulated or when flux noise is considered. In this case, application of the existing quantization procedure can lead to inconsistencies. To resolve these, we generalize circuit quantization to incorporate time-dependent external flux.

Figure 1

(a) Asymmetric SQUID formed by two Josephson junctions. ${\mathrm{\Phi }}_{\text{l}}$ and ${\mathrm{\Phi }}_{\text{r}}$ are branch flux variables for the left and right junction. The node flux $\varphi$ is associated with the upper node, while the lower node is chosen to be the ground node. An external flux ${\mathrm{\Phi }}_{\text{e}}$ threads the loop. (b) and (c) depict the two equivalent choices of spanning trees (solid lines) and closure branches (dashed lines).

Figure 2

The overlap between two states at different times is not conserved under a time-dependent unitary transformation, since $\langle \psi \left(0\right)\left|U\left(0\right)U^{†}\left(t\right)\right|\psi \left(t\right)\rangle \ne \langle \psi \left(0\right)|\psi \left(t\right)\rangle$ in general.

Figure 3

The fluxonium qubit is formed by a Josephson junction and a superinductor with corresponding branch variables ${\mathrm{\Phi }}_{\text{J}}$ and ${\mathrm{\Phi }}_{\text{L}}$. An external flux ${\mathrm{\Phi }}_{\text{e}}$ threads the loop.

Figure 4

Multiloop circuit formed by two SQUID loops with one shared junction. ${\mathrm{\Phi }}_{\text{1}}$, ${\mathrm{\Phi }}_{\text{2}}$, and ${\mathrm{\Phi }}_{\text{3}}$ are branch flux variables for the three Josephson junctions. External fluxes ${\mathrm{\Phi }}_{\text{e}}^1$ and ${\mathrm{\Phi }}_{\text{e}}^2$ thread the left and right loops.